Don Kirkham

Gaseous diffusion equations for porous materials

Commonly used equations of gaseous transport by diffusion are examined and the two parameters required for each equation are tabulated from the literature. The advantages and limitations of the commonly used linear —D/Do = a(S−b) — and curvilinear — D/Do = KSm— equations are considered and a new diffusion equation is proposed to combine the advantages of the previous equations. This new equation, like the others, requires only two parameters. It takes the form D/Do = [(S−u)/(1−u)]v and may serve as a basis for studying and more accurately modeling gas transport in porous media.

Soil Water Evaporation Suppression By Sand Mulches

This paper reports experiments performed to study sand as a soil mulch. The objective was to determine the comparative effectiveness of 0-, 2-, and 6-cm-thick covering sand layers in suppressing evaporation from columns of soil. Measurements were made by using potential evaporation rates of 1.1 and 0.55 cm/d. In addition to evaporation, soil water distribution with depth was measured for the different sand-cover treatments. Five treatments were studied: check (no sand mulch), 6 cm of coarse sand (C6), 6 cm of fine sand (F6), 2 cm of coarse sand (C2), and 2 cm of fine sand (F2). After 35 d of experiment, the cumulative evaporations for the check, C6, F6, C2, and F2 treatments were measured as 6.79, 1.50, 1.55, 3.76, and 4.62 cm of water, respectively, at a potential evaporation of 1.1 cm/d and, for the potential evaporation of 0.55 cm/d, was correspondingly 6.68, 0.95, 1.21, 2.71, and 4.28 cm. These sets of numbers show that there was marked evaporation reduction for the sand mulches with respect to bare soil (check). The 6-cm sand mulches were the most effective evaporation suppressors. For equal mulch thickness, coarse sand was only slightly more effective than fine sand. Results from soil water distributions with depth for the various treatments also indicated that the sand mulches were effective in conserving soil water against evaporation losses. The mulches were effective in this order: C6 > F6 > C2 > F2.

Potential Flow into Circumferential Openings in Drain Tubes

A theoretical analysis of the effect of the spaces between drain tube units as used in the artificial drainage of soil is given. The problem is one of potential flow; therefore, the results are applicable to heat flow, etc. The basic problem solved is that for axially symmetric flow from an external cylindrical boundary at constant potential to a series of equal, equally spaced openings at a lower potential, all located axially on, and comprising a part of, the otherwise impervious drain tube. The radii of the open sections and impermeable sections of the drain tube are equal. The basic problem is extended to obtain the solution to the practical problem—the seepage of ground water into drain tubes beneath a horizontal water table. The exact solution of the basic problem is not suitable for numerical work. Accordingly, approximate solutions of specified uncertainty are derived and are utilized for tabulation of numerical results. As an example, the analysis shows, in the case of 6 in. diameter drain tubes ...

Seepage of leaching water into drainage ditches of unequal water level heights

Abstract The theory is developed, and numerical examples are worked out for the seepage of ponded water into equally spaced ditches in homogeneous soil overlying a barrier. Two general cases are solved: (a) unequal depth of water in the drainage ditches and (b) equal depth. For both (a) and (b) the problem is solved for zero depth of ponded water and for finite depth of ponded water; these solutions are quite different. Curves are supplied showing the surface inflow seepage velocity as it varies with distance from a ditch. Near ditches the seepage inflow rate is very high compared with that between ditches, which helps explain why, many times, crops do better when near ditches than when removed from them in reclaimed saline soil.

Solute travel times to wells in single or multiple layered aquifers

Abstract Approximate travel times are given for a solute to reach a well from an injection point, for a well of finite radius screened throughout an aquifer resting on an impervious sole where: (1) there is a constant head at the outer boundary of the flow region; and (2) there is no flow across the outer boundary (both models were discussed earlier). In both cases, the time of travel for a solute entering the top of the aquifer at a point rk satisfies approximately a logarithmic relationship as follows: In rk = A ln t + B where t is the time of travel from the point rk to the well, and A and B are constants depending on the problem considered. This linear relationship holds only up to a certain distance of injection from the well. This distance varies with the problem considered.

Transport of nitrate and gaseous denitrification in soil columns during leaching

Abstract Leaching of nitrate with first-order denitrification in a soil column of finite length has been theoretically analyzed. Transport equations and boundary conditions describing the movement of nitrate and denitrification, by chemical and/or microbiological means, in a soil column were set up and solved for continuous application of nitrate at the soil surface. From the mathematical model presented, relative concentration profiles can be determined for any time and known values of the diffusion coefficient and average solute velocity. The validity of this model has been supported by comparison of the results with experimental data from the literature.

Flow patterns of steady rainfall seeping through bedded land or a hillside with a barrier at great depth

Abstract The problem of the steady seepage of rainfall or other recharge through bedded fields underlain by an impermeable barrier at a great depth is solved for finite depths of water in the bedding furrows. The problem solution can apply to soil bedding, to hillside seepage as to a river, and to closely spaced furrows and ridges as in row cropping. Two basic geometries are considered, a rectilinear (tent shape) bedding and an elliptic (ellipse shape) bedding for homogeneous soil. It is assumed that Laplaces equation is valid. A potential function is obtained by use of a modified Gram-Schmidt method. The rainfall rate needed to keep the bedding saturated, the average quantity per unit area per unit time of water flowing through the soil, and the ratio of the latter to the former are calculated. By changing the shape of the soil surface from rectilinear to elliptic, the ratio for geometries considered greatly increases, about 400%. Flow nets for several soil-bedding geometries are presented. From these nets, it is seen that water enters the soil upslope above a critical point and, after flowing through the soil, resurfaces again downslope and enters surface runoff. The critical point is located farther downslope for elliptic bedding than for rectilinear and also is located farther downslope for a finite depth of barrier than for an infinite depth. This in-and-out seepage of water illustrates how soluble material may be removed from the soil and be added to surface runoff. The flow nets also show how seepage spots on a hillside may develop and how a type of base flow may occur. When rain or other applied recharge ceases, the soil water itself, draining from the soil surface region, may be considered to supply the recharge in quasi-steady flow for a limited time. This quasi-steady recharge drains downward over a larger percentage of the surface area of elliptic bedding than of rectilinear and thus indicates that elliptic is better than rectilinear for removal of soil water.

Use of radiation equipment for plow-layer density and moisture.

Radiation probes were found to be quicker and give approximately the same precision as core sampling for determinations of the approximate plow-layer density and moisture content of soil. Results of two field experiments are reported.

Potential and Capacity of Concentric Coaxial Capped Cylinders

An exact solution is obtained for the potential in the space between two finite concentric coaxial right circular capped cylinders; and from the potentials an exact solution of the capacity is found. A set of infinite equations is involved but detailed numerical calculations, made for nine different geometries, show that if the ratio of height of the outer cylinder to diameter of the inner cylinder is less than about unity only the first eight equations and the first eight unknowns in them need be considered for obtaining three figure accuracy of the capacity. The theory is correct when the inner cylinder shrinks to a disk and also when the radius of the outer cylinder goes to infinity. Thus the theory yields the capacity for a horizontal disk midway between infinite conducting planes. When the planes become infinitesimally close to the disk the problem becomes two‐dimensional, and the three‐dimensional capacity expression goes over to an expression obtained by a conformal transformation. A table of coefficients for 1st, 2nd, 3rd, 4th, 6th, and 8th order approximations is given for computing quantities of interest and to show the rapidity of convergence of results for the nine geometries considered. For one of the cases the equipotentials and lines of force are computed and diagrammed.

The stream function of potential theory for a dual‐pipe subirrigation‐drainage system

An exact mathematical solution to Laplaces equation is presented for appropriate boundary conditions associated with the problem of dual-pipe subirrigation and drainage. The solution can be used to determine a flow net within the groundwater flow region and the associated water table shape. The solution is general. The effects of several hydraulic and geometrical parameters on the groundwater system, such as thickness of saturated zone, position of subirrigation and drainage pipes, heads in the subirrigation and drainage pipes, crop evapotranspiration, fraction of inflowing subirrigation water that exits through the drains, and the aquifer hydraulic conductivity system are evaluated. Calculations are presented showing how pipe spacing affects the shape of the water table. For example, with hydraulic conductivity of 10 m/d, evapotranspiration of 0.01 m/d, drainpipe radius of 0.05 m, and subirrigation pipe radius of 0.0375 m, calculations show that the maximum water table elevation for a pipe spacing of 40 m is 0.64 m greater than for a pipe spacing of 16 m when 40% of the input subirrigation water volume is being removed from the system by drainage. Finally, the general mathematical solution can be used to predict chemical movement as well as water flow through the system.